On growth characteristics of operator-valued functions

Abstract

In the work Liouville theorem and the concept of order and type of entire function are generalized to the case of operator-valued function with values in the space $\mathrm{Lec}(\mathbf{H}_1,\mathbf{H})$ of all linear continuous operators acting from a locally convex space $\mathbf{H}_1$ to a locally convex space $\mathbf{H}$ with equicontinuous bornology. We find the formulae expressing the order and type of operator-valued function in terms of characteristics of the sequence of coefficients. Some properties of order and type of operator-valued function are established.